Optimization of Biomass Composition Explains Microbial Growth-Stoichiometry RelationshipsAuthor(s): Oskar Franklin, Edward K. Hall, Christina Kaiser, Tom J. Battin, Andreas RichterSource: The American Naturalist, Vol. 177, No. 2 (February 2011), pp. E29-E42Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/10.1086/657684 .Accessed: 01/02/2011 10:04

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vol. 177, no. 2 the american naturalist february 2011

E-Article

Optimization of Biomass Composition Explains Microbial

Growth-Stoichiometry Relationships

Oskar Franklin,1,* Edward K. Hall,2 Christina Kaiser,3 Tom J. Battin,2 and Andreas Richter3

1. International Institute for Applied Systems Analysis (IIASA), A-2361 Laxenburg, Austria; 2. Department of Limnology, University ofVienna, Althanstrasse 14, A-1090, Vienna, Austria; 3. Department of Chemical Ecology, University of Vienna, Althanstrasse 14, A-1090,Vienna, Austria

Submitted June 1, 2010; Accepted September 17, 2010; Electronically published January 12, 2011

abstract: Integrating microbial physiology and biomass stoichi-ometry opens far-reaching possibilities for linking microbial dynam-ics to ecosystem processes. For example, the growth-rate hypothesis(GRH) predicts positive correlations among growth rate, RNA con-tent, and biomass phosphorus (P) content. Such relationships havebeen used to infer patterns of microbial activity, resource availability,and nutrient recycling in ecosystems. However, for microorganismsit is unclear under which resource conditions the GRH applies. Wedeveloped a model to test whether the response of microbial biomassstoichiometry to variable resource stoichiometry can be explainedby a trade-off among cellular components that maximizes growth.The results show mechanistically why the GRH is valid under Plimitation but not under N limitation. We also show why variabilityof growth rate–biomass stoichiometry relationships is lower underP limitation than under N or C limitation. These theoretical resultsare supported by experimental data on macromolecular composition(RNA, DNA, and protein) and biomass stoichiometry from two dif-ferent bacteria. In addition, compared to a model with strictly ho-meostatic biomass, the optimization mechanism we suggest resultsin increased microbial N and P mineralization during organic-matterdecomposition. Therefore, this mechanism may also have importantimplications for our understanding of nutrient cycling in ecosystems.

Keywords: optimization model, mineralization, growth-rate hypoth-esis, RNA, biomass stoichiometry, microbial physiology.

Introduction

Ecological stoichiometry provides a powerful tool for in-tegrating microbial physiology and stoichiometry withecosystem processes. For example, the growth-rate hy-pothesis (GRH) predicts that growth rate increases withphosphorous concentration through changes in RNA con-tent (Sterner 1995), which has been observed in organismsfrom microbes (Makino et al. 2003) to humans (Elser et

* Corresponding author; e-mail: franklin@iiasa.ac.at.

Am. Nat. 2011. Vol. 177, pp. E29–E42. � 2011 by The University of Chicago.

0003-0147/2011/17702-52207$15.00. All rights reserved.

DOI: 10.1086/657684

al. 2007). Recent theoretical advances show that the GRH,coupled with the metabolic theory of ecology, can explainstoichiometric patterns across organism types (Allen andGillooly 2009). The GRH also underlies the experimentalmethods used to estimate microbial activity on the basisof ribosomal RNA (rRNA) content (Leser et al. 1995).However, these methods and theories should be inter-preted with care because observations (Flardh et al. 1992;Binder and Liu 1998; Elser et al. 2003) show that the GRHis not universally valid, especially in microorganisms. Therelationship between bacterial biomass stoichiometry andgrowth rate can vary within the same species (Chrzanowskiand Kyle 1996; Makino et al. 2003; Chrzanowski andGrover 2008) and depend on which nutrient is limitinggrowth (Sepers 1986). The mechanisms behind this var-iability are not yet well understood. Moreover, it has beenproposed that a general lack of theory limits progress inthe field of microbial ecology (Prosser et al. 2007). Thus,developing theory and models is an important step inelucidating the mechanisms behind the variable stoichi-ometry of microorganisms and its implications for nutri-ent cycling in ecosystems.

The response of microbial growth to variation in re-source stoichiometry has commonly been modeled withcellular quotas, that is, the Droop model (Droop 1968;Thingstad 1987). A phytoplankton model by Klausmeieret al. (2004), also based on quotas, took this approach onestep further by adding mechanistic detail linking the min-imum quotas to structural composition in terms of re-source acquisition and growth machinery. More impor-tantly, this model employed an optimization strategy oftrade-offs between resource acquisition machinery andgrowth machinery to control biomass composition. Op-timality assumptions are attractive because they providean ecological (and/or evolutionary) rationale for modelbehavior. Whereas optimal-biomass-allocation principleshave been widely used to explain the response of plantsand animals to resource availability (e.g., Kozłowski et al.

E30 The American Naturalist

Figure 1: Model structure. a, Model components and fluxes (processes). b, Effect of biomass partitioning on cell processes for a growth-limiting element. Arrows indicate positive (plus sign), negative (minus sign), or variable (plus-minus) effect. Biomass composition (u, g, andz) has direct functional effects on growth and uptake and an indirect effect, through biomass element concentration, that affects nutrientmetabolite consumption per unit of biomass growth (nutrient-use efficiency). For clarity, excretion (which is negligible for a limiting element)and respiration (which is constant) are not shown.

2004; Franklin et al. 2009), the few such studies that havebeen conducted for microorganisms are (to our knowl-edge) restricted to phytoplankton (e.g., Wirtz 2002; Klaus-meier et al. 2004). For bacteria, proteome studies suggesttrade-offs between growth and nutrient acquisition (e.g.,Raman et al. 2005). However, the hypothesis that optimalallocation of cellular machinery explains the response ofgrowth rate and biomass stoichiometry to resource stoi-chiometry in heterotrophic microorganisms has yet to betested.

Here we take the optimization approach (Klausmeier etal. 2004) one step further in terms of mechanistic detailby explicitly modeling the growth process as a function ofinternal resource pools and cellular growth and uptakemachinery, each with a fixed elemental composition. Wehypothesize an optimal partitioning of biomass amonggrowth machinery, uptake machinery, and other structuralbiomass. We then evaluate how optimization among thesecomponents affects biomass composition and stoichiom-etry under variable resource stoichiometry. Specifically, weshow (1) that optimization of biomass composition thatmaximizes specific growth rate explains variability in bac-terial biomass stoichiometry, (2) that resource stoichi-ometry strongly influences macromolecular composition(e.g., RNA content) so that the GRH is valid under Plimitation but not under N limitation, and (3) that thepresence of an optimization mechanism increases P andN recycling under C-limited bacterial growth, comparedto that when biomass is strictly homeostatic.

Theory and Model

Model Structure and the Optimal-Biomass-Composition Hypotheses

In our model, structural biomass (fig. 1; table 1) is dividedinto cellular compartments with specific functions: (1)baseline biomass, denoted z (DNA, cellular membrane, cellwall, essential proteins), (2) growth machinery g (ribo-somes, RNA), and (3) uptake machinery u (transmem-brane proteins). In addition to structural biomass, we con-sider internal C, N, and P metabolite pools that are usedfor growth and respiration. All processes and variables(described below; table 2) are defined on a per-biomassbasis, which allows us to not explicitly consider effects ofcell volume or density changes. Unless indicated, biomassrefers to the structural biomass only (excluding the inter-nal metabolite pools).

For a given resource level (s), the relative partitioningof biomass among the three cell compartments (uptakemachinery u, growth machinery g, and baseline biomassz) is adjusted to maximize the specific growth rate G.Specifically, the optimal cell composition depends on thebalance between growth and uptake capacity and is furtherinfluenced by the elemental (C, N, P) demand for theconstruction of the selected biomass composition (fig. 1b).The fraction of baseline biomass z is, however, assumedto always be greater than a minimum value zmin to maintainessential functions other than growth and uptake. Ad-justments of biomass composition, for example, amount

Bacterial Stoichiometry E31

Table 1: Composition of cell compartments

Cell compartment Symbol Composition Protein C N P

Baseline biomass z Cell wall, membranes, DNA (15, 17a) 62, NDa 43 14.9, 13.7a 1.0, 1.0a

Uptake machinery ub Protein 100 46 17 0Growth machinery gb Ribosomal protein (33), RNA (67)c 33 37 15.3 5.87C in metabolites pC Carbohydrates (50), lipids (50) 0 54.2d 0 0N in metabolites pN NH4 (50), amino acids (50) 0 23 47.4 0P in metabolites pP NaPO4 (50), NaPO3 (50) 0 0 0 28

Note: All values are in percent mass. Unless indicated, C, N, and P percentages in macromolecules and cell compartments are taken

from Sterner and Elser (2002).a Estimated from data for Pectobacterium carotovorum and Escherichia coli, respectively. ND p not determined.b Mathematically, u and g are the uptake and growth fractions, respectively, of the nonbaseline biomass ( ).1 � zc Assuming that 85% of RNA is in ribosomes and that ribosome RNA / protein p 1.8 for prokaryotes (Sterner and Elser 2002).d From Vrede et al. (2004).

of RNA, have been shown to occur rapidly in response toenvironmental changes and in experiments (Kerkhof andKemp 1999; Ferenci 2007). Thus, we assume that biomasscomposition is in dynamic equilibrium with the environ-mental conditions, although we also evaluate under whichresource conditions the biomass composition may deviatefrom the optimal equilibrium-based predictions.

Cellular Compartments and Stoichiometry

The elemental composition of each cellular compartment(table 1) is based on values derived from the literatureand original research presented in this study (see below).Only the elements C, N, and P are explicitly modeled,whereas all other atoms in biomass and metabolites areimplicit. Specifically, the proportion of element X in bio-mass (bX, where X denotes C, N, or P) is the product ofthe relative amount of each cellular compartment and itselemental content (Xg, Xu, Xz):

b p z(X ) � (1 � z)(gX � uX )X z g u

p z(X ) � (1 � z)[gX � (1 � g)X ]. (1)z g u

In equation (1), g and u are defined as fractions of thenonbaseline biomass, so that . Resources areu p 1 � gtaken up from the environment and stored in metabolitepools that are represented by their C, N, and P contents,although the actual chemical forms of the resource pools(metabolites), for example, carbohydrates (C), amino acids(N, C), and phosphates (P), are variable. Nutrients aretapped from the metabolite pools (p) for growth of newbiomass (G) and respiration (R) or are excreted (e.g., over-flow metabolism E). Metabolite pools are assumed to bein dynamic equilibrium, so that their size depends on thebalance between nutrient uptake (U) and use:

dpX p U � Gb � E p 0,X X Xdt

dpC p U � Gb � E � R p 0, (2)C C Cdt

where X represents N or P.

Cell Processes

Growth. The growth model is based on the synthesizing-unit (SU) concept, which is based on the microscopicinteractions of growth machinery and different nutrients(app. A; Kooijman 1998, 2001). In our framework, the SUconcept leads to a growth equation (eq. [3]) in which Gincreases with the ratio of metabolite concentration (pX)to the demand of the same nutrient for biomass growth(controlled by bX) until G approaches its maximum Gmax:

db 1p G

dt b�1

G b b b p pmax C N P C Np G 1 � � � � �Zmax { [ ( )e p p p b bg C N P C N

�1 �1

p p p pC P P N� � � � (3)( ) ( )b b b bC P P N

�1

p p pC N P� � � .( ) ]}b b bC N P

In equation (3), eg is the initial efficiency of the SU, thatis, the growth per metabolite availability at low metaboliteconcentration (pX), and Gmax (maximum G) is proportionalto the amount of growth machinery g ( ) and its max-1 � zimum capacity of biomass synthesis (fG), which is a func-tion of the maximal translational activity of the ribosomes(Jackson et al. 2008):

G p f (1 � z)g. (4)max G

E32 The American Naturalist

Table 2: Variables and parameters

Symbol Unita Valuesb Description

Variables:bX g X g B�1 ... Fraction of element X in biomassEX g X g B�1 h�1 ... Specific rate of excretion of element Xg ... ... Fraction of nonbaseline biomass in growth machineryG h�1 ... Specific growth ratepX g X g B�1 ... Metabolite pool of element XUX g X g B�1 h�1 ... Resource uptake of element Xu ... ... Fraction of nonbaseline biomass in uptake machinerysX g X g B�1 h�1 ... External resource level of element Xz ... ... Fraction of baseline biomass of total structural biomass

Parameters:eg ... P.c.: .38; E.c.: 56 Maximum efficiency of the growth machineryfE ... .01 Excretion rate factorfG h�1 P.c.: .93; E.c.: 4.2 Maximal synthetic capacity of the growth machineryplim X g X g B�1 P.c.: .43, .10, .021;

E.c.: .42, .029, .0085Theoretical maximum metabolite pools of C, N, and P, respectively

r g C g B�1 .02c Baseline maintenance respirationy ... .5d Growth efficiency: C growth per C used in the growth processzmin ... P.c.: .56; E.c.: .19 Minimum fraction of baseline biomass of total structural biomass

a B p structural biomass (biomass excluding the nutrient metabolite pools). X p any of the elements C, N, or P.b P.c. and E.c. p Pectobacterium carotovorum and Escherichia coli, respectively. All parameter values are estimated from our data except as noted below.c From Tannler et al. (2008).d From Cajal-Medrano and Maske (1999).

It is commonly assumed that a single nutrient limitsgrowth (cf. Liebig’s law). In our model, this means thatfor the limiting nutrient, the ratio of the metabolite pool(pX) to biomass concentration (bX) is much smaller thanthat for the nonlimiting nutrients. Under these circum-stances, equation (3) can be approximated by equation(5), which more clearly illustrates the interaction of Gmax

and each metabolite pool, that is, why small metabolitepools limit growth whereas Gmax limits growth when me-tabolite pools are large:

GmaxG p . (5)1 � (G /e )(b /p )max g X X

In equation [5], the subscript X refers to the most-limitingnutrient.

Uptake. Similar to the growth dynamics limited by Gmax,resource uptake (U; eq. [6]) is limited by uptake capacity(Umax) at a high external resource level (s), whereas it islimited by s at low s (cf. Brandt et al. 2004; i.e., a JacobMonod– or Michaelis-Menten-type functional form):

U smax X XU p . (6)X U � smax X X

Here Umax is a function of uptake machinery (u), as definedbelow. Because we consider external resources solely interms of their total effects on potential rate of element

uptake, we can neglect underlying details of the uptakeresponse to s, such as effects of different resource types.Instead, s represents the total effect of resource level onuptake, that is, sX is the uptake of element X when thelimitation by Umax is removed and U is completely resourcelimited. Uptake of a nonlimiting element is approximatedby in our simulations.U p Umax

Excretion and Respiration. To avoid having any elementreach an unrealistic or deleterious concentration, there isan upper limit to the size of each internal nutrient pool(p; Russell and Cook 1995). If p approaches its upper limit(plim), then nutrients are excreted or rejected (for P andN) or respired (for C, overflow respiration) according to

f pE XE p (7)X p � plim X X

(fig. B1). In equation (7), fE controls how quickly E in-creases with p, and it is assumed to be low ( ) sof p 0.01E

that E is minimal below .p r p lim

Anabolic respiration associated with the construction ofbiomass (RG) is proportional to growth (G); that is,

, where y is the growth efficiency (y pR p [1/(y � 1)]GG

biomass C growth / total C used in the growth process).Although y may be variable among species and for dif-ferent environmental conditions, differences in y have only

Bacterial Stoichiometry E33

quantitative effects under C limitation but no qualitativeeffects on our results. Thus, for simplicity, we assume agrowth efficiency of (Cajal-Medrano and Maskey p 0.51999). Specific maintenance respiration (Rm) is assumedto be constant for each species (Pirt 1982) but variesamong species. A linear relationship between maximumG and specific maintenance rate (Rm) has been observedacross species (van Bodegom 2007); this can be explainedby the energetic costs of protein synthesis machinery (hereg) increasing with its translation speed (here fG; Dethlefsenand Schmidt 2007). In our model, this relationship cor-responds to Rm proportional to fG, where (Tann-r p 0.02ler et al. 2008) is the baseline respiration:

R p rf . (8)m G

Balancing Capacities for Nutrient Uptake and Use. Uptakecapacity and growth capacity are linked to their respectiveproportions (g and u) of the nonbaseline biomass (1 �) through a trade-off ( ), so that both capacitiesz u p 1 � g

cannot be maximized simultaneously. Furthermore, giventhe benefits of the ability to utilize and buffer variationsin resource supply (e.g., Thomas and O’Shea 2005), max-imum uptake capacity should be higher than maximumgrowth capacity. Specifically, we assume that a referenceuptake capacity, given by , suffices to matchu p u p 0.50

maximum growth capacity (Umax at equalsg p u p 0.5GmaxbX at ) for each nutrient, which leads to equationg p 1(9). However, as long as the maximum uptake capacity isnot smaller than maximum growth capacity, our resultsare not sensitive to this assumption.

f bG XU p u (1 � z),max X u 0

f b (1 � z ) u (1 � z)G C minU p � rf , (9)max C G[ ]y (1 � z )umin 0

for . In equation (9), bX (eq. [1]) is evaluatedX p N or Pat and .g p u p 0.5 z p z0 min

Model Evaluation

Solving for Optimal Biomass Composition. To evaluate themodel, G (eq. [3]) was maximized with respect to biomasscomposition (g and z) for each substrate level (s), underthe constraint . We numerically solved for s toz 1 zmin

obtain optimal g and z as a function of G. In each case,single-element limitation was modeled; that is, only oneresource element at a time affected G, while the effects ofthe other resources were fixed by setting their uptake rate

(see “Uptake”). In addition, we evaluated theU p Umax

robustness of the optimal biomass composition in terms

of the probability for suboptimal values of g and z. Forexample, because of rapid fluctuations in resource level,cells may not always be in dynamic equilibrium with theenvironment and may therefore deviate from the modeledoptimal composition, which would lead to variationaround the optimal G-to-g and G-to-z relationships. Therange of this variability should be larger the less sensitiveG is to deviation of g and z from their respective optima.Thus, as a measure of potential variability in g and z, wecalculated how far each parameter can be from the optimawithout reducing G by more than 5%.

Model Testing and Evaluation. To test the most centralassumption in the model, optimal biomass partitioning,we compared model predictions and data for concentra-tions of macromolecules specific to two of the three bio-mass compartments. The data came from a chemostatexperiment using Escherichia coli (Makino et al. 2003) anda batch culture experiment using Pectobacterium caroto-vorum (app. C; Keiblinger et al. 2010). RNA was used asan index of growth machinery (as 85% of cellular RNAis rRNA; table 1), and DNA was used as an index ofbaseline biomass. To minimize the effect of changes innonstructural components, that is, metabolite pools, weanalyzed RNA and DNA relative to protein content (forP. carotovorum), which is not affected by metabolite pools.When protein data were not available (for E. coli), theywere replaced by total N content. The model was fit-ted (see method below) to measured RNA : protein,DNA : protein, and P metabolite pool : protein for P. caro-tovorum and to RNA : N and DNA : N for E. coli. For thetesting of metabolite pool dynamics we chose P, becausewe can more easily separate metabolites and structuralcomponents—that is, amounts in RNA (fixed proportionof growth machinery) and in baseline biomass (constantunder P limitation; table 1)—for P than for C and N,which occur in all cell compartments and in more thanone metabolite pool. For P. carotovorum, the limiting nu-trient was identified for each treatment on the basis ofmeasured dynamics of metabolite pools and storage (poly-phosphate and carbohydrates) in response to growth rate(app. C). For E. coli, we relied on limitations identified inthe original study (Makino et al. 2003). Finally, we eval-uated the consequences of our optimization hypothesis fornutrient recycling during decomposition by simulating theeffect of declining C availability, which is ubiquitous dur-ing organic-matter decomposition.

Parameterization. Although the elemental composition ofthe macromolecules and the macromolecular compositionof the cellular compartments are constrained (Sterner andElser 2002), some components vary among species or aredifficult to estimate accurately from literature data. Thus,

E34 The American Naturalist

we estimated species-specific values for minimum baselinebiomass (zmin) and its N, protein, and DNA contents fromdata for each species. For the process-related parameters,we estimated species-specific values for the maximum ca-pacity (fG) and efficiency (eg) of the growth machinery andthe maximum metabolite pool sizes (plim X). Best-fit pa-rameters (maximum likelihood of yielding the measureddata) were estimated via Markov chain Monte Carlo (e.g.,Gelman et al. 2004), which has the advantage, comparedto standard optimum-seeking methods, of minimizing therisk of selecting a local but not global optimum for theparameter values. For all calculations we used MathCad(ver. 13) software (code available in a zip file).

Results: Model Performance and Behavior

Modeled and Measured Relationships between GrowthRate and Macromolecular Biomass Composition

In order to evaluate the presence of the hypothesized op-timization mechanism, we compared measurements withmodel predictions (optimal and potential suboptimalranges of variability) of covariation of biomass composi-tion and specific growth rate induced by variation in re-source level and resource C : N : P stoichiometry. Specif-ically, we tested for RNA : protein, DNA : protein, and Pmetabolite pools : protein ratios under N and P limitationfor Pectobacterium carotovorum. For Escherichia coli, pro-tein data were not available, so we tested for RNA : N andDNA : N under C and P limitation.

For P. carotovorum, the modeled biomass compositionversus growth rate G differed significantly between N-lim-ited growth and P-limited growth (fig. 2). Under P lim-itation, modeled RNA : protein increased linearly with G,in accordance with the growth-rate hypothesis (GRH).However, under N limitation, the modeled RNA : proteinrelationship with G was nonlinear and clearly defied theGRH. Also, the modeled DNA : protein ratio differed be-tween P and N limitation. Whereas this ratio did not varywith G under P limitation, it increased with declining Gunder N limitation, resulting from a relative increase inbaseline biomass (z, which includes DNA) and a relativereduction of both growth and uptake machinery. Thesemodel predictions of optimal biomass composition wereconsistent with the observed trends in RNA : DNA : pro-tein ratios versus G, capturing the differences in theserelationships between P and N limitation (fig. 2). In ad-dition, most of the observed variability of the responsevariables was within the modeled range of potential var-iability (fig. 2, thin lines), although under P limitation someobserved variability in DNA : protein could not be readilyexplained by the model. For E. coli, the model predictedlinearly increasing RNA : N versus G under P limitationand a similar relationship, with a smaller slope, under C

limitation over the range of observed G. For optimalDNA : N, no effect of G or difference between C and Plimitation was predicted by the model. However, for bothDNA : N and RNA : N the model suggests a higher po-tential variability under C limitation than under P limi-tation. These modeled trends for optimal biomass com-position were again consistent with observations (fig. 2).

The modeled P metabolite pool in P. carotovorum dif-fered significantly between P-limited and non-P-limitedconditions, resulting in a much lower pool under P lim-itation than under N limitation (fig. 3). In addition, underP limitation the P metabolite pool increased strongly withincreasing G, whereas such a monotonic relationship wasnot present under N limitation. The model implies thatthis difference reflects a general difference between themetabolite pool dynamics of limiting and nonlimiting nu-trients. For a limiting nutrient, the metabolite pool (p)increases with G because p is a dominant control of G (fig.1b; eq. [5]). In contrast, nonlimiting nutrients can accu-mulate in metabolite pools without strong effects on G.Instead, nonlimiting metabolites are passively controlledby the flux balance of the cell; for example, they are re-duced through increased use when G increases at constantuptake capacity. This difference between limiting and non-limiting conditions was confirmed by the measured trendsin P metabolites in P. carotovorum (fig. 3).

The differences in estimated model parameters (themodel interpretation of the empirical data) between thetwo bacterial species imply that E. coli reaches a higher Gthan does P. carotovorum because of the higher capacity(fG) and higher efficiency (eg) of its growth machinery. Thedifference in G is further enhanced by E. coli’s lower min-imum requirement for baseline biomass (zmin). The modelalso suggests that the species differed in their maximummetabolite pools for P and N, although we were not ableto evaluate this using the available empirical data.

In summary, the model predicted trends in the rela-tionships among biomass G, RNA, DNA, and protein andhow those relationships differed under C, N, and P lim-itation, in agreement with observations. This result is con-sistent with an optimization of biomass composition tomaximize specific growth rate in response to resource lev-els. In addition, the agreement between modeled and mea-sured dynamics of the metabolite pools was consistent withthe growth and uptake mechanisms proposed in themodel.

Implications for a Microbial Process, Nutrient Recycling

During microbial decomposition of organic matter, re-source C : N and C : P ratios gradually decline because ofthe loss of C through microbial respiration. We evaluatedthe effect of our optimization hypothesis on this process

Bacterial Stoichiometry E35

Figure 2: Relationships between biomass composition ratios and specific growth rate (G) induced by variation in external resource levelsfor two species of bacteria. Modeled (lines) and measured (symbols) results for growth under different single-element limitations: N limitation(dashed lines, circles), P limitation (solid lines, triangles), and C limitation (dash-dotted lines, squares). The optimal state (thick middle line)and intervals of near-optimal states (95% optimal; upper and lower thin lines) were modeled. The distance between the thin lines representsthe range of near-optimal values. For Pectobacterium carotovorum, and 0.10 for RNA : protein and DNA : protein, respectively,2r p 0.79under P limitation and 0.43 and 0.63, respectively, under N limitation. For Escherichia coli, and 0.31 for RNA : N and DNA : N,2r p 0.86respectively, under P limitation, and 0.57 and 0.086, respectively, under C limitation.

for a bacterium (P. carotovorum) by model simulation ofdeclining resource C level at two levels of fixed resourceN availability and constant resource P (fig. 4).

Compared to the case when biomass was homeostatic,optimization of biomass composition led to a small in-crease in growth rate and a slight change in biomassC : N but a significant change in biomass N : P when Climitation set in. More importantly, modeled recycling(mineralization) of both N and P was increased becauseof the optimization. This is explained by the shift from alow proportion of uptake machinery (u) under N limi-tation to higher u under C limitation (mechanism ex-plained below). This shift increases uptake and thereforealso excretion of excess nonlimiting elements, that is, P

and N. If the resource N level is reduced, the resourceC : N ratio where the shift to C limitation occurs is in-creased. This effect of resource N level is due to the effectof maintenance respiration that causes a growth-indepen-dent C-use term, which means that total C use does notdecline in proportion to N use as resource N level declines.

Discussion

Mechanisms Regulating the Contrasting BiomassComposition among P, N, and C Limitation

In both the model and the empirical studies, growth rate(G) scaled consistently with RNA and biomass P content

E36 The American Naturalist

Figure 3: Relationships between metabolite pool phosphorus : bio-mass protein ratio and specific growth rate (G) induced by variationin external resource levels. The growth-limiting elements were ni-trogen (N) and phosphorus (P; symbols and lines as in fig. 2). Thelower thin solid line for P limitation (near-optimal value) coincideswith the thick solid line (optimal value). Measured P metabolite poolswere calculated from the P budget as , whereP � P � Ptotal RNA z

. Pz non-DNA was the non-DNA P content in base-P p P � Pz DNA z non-DNA

line biomass (z) and was estimated as 0.28% by fitting the y-interceptof the metabolite pool versus G under P limitation to 0. For Plimitation, one outlier, at , for which the estimated PG p 0.065metabolite pool was !0 was removed from the analysis. 2r p 0.32and 0.87 under N and P limitation, respectively.

under P limitation and, at higher growth rates, under Climitation, but not under N limitation. To understandthese differences, it is important to understand the twointeracting mechanisms through which biomass partition-ing maximizes growth rate in our model. First, changesin proportions of growth and uptake machinery (g and u,respectively) cause a functional trade-off between growthand uptake capacity. Second, the resultant changes in bio-mass composition affect the demand for (use of) the lim-iting element per biomass constructed; that is, a nutrient-use efficiency (NUE) effect emerges. This effect determinesthe rate at which the metabolite pool is depleted for agiven growth rate (fig. 1b). While the functional trade-offalways supports the GRH, the NUE effect does not. TheNUE effect is important at low resource levels, where itresults in biomass patterns that are consistent with theGRH under P limitation but not under N limitation.Therefore, resource level ultimately determines whichmechanism controls the relationship between biomassstoichiometry and growth.

The optimization of the functional trade-off acts equallyfor all limiting elements, by favoring growth machinery at

the expense of uptake machinery as resource level in-creases. At a high resource level, this optimization of func-tional capacity is the dominant mechanism controllingbiomass composition. Under these conditions, the positiverelationships between RNA, G, and biomass P may emergeindependently of which element is limiting, consistent withthe GRH. Baseline biomass z will always be at its minimumvalue because it does not contribute to either uptake orgrowth capacity.

At declining resource levels, the effects of uptake andgrowth capacity decline because uptake is controlled byresource level rather than uptake capacity (fig. 1b; eq. [6]).As a consequence, the relative importance of the NUEeffect increases and biomass is allocated toward the com-partment with the lowest concentration of the limitingnutrient. For example, under P limitation, because uptakemachinery is lowest and growth machinery highest in P,the NUE effect reinforces the functional trade-off that re-duces growth machinery as available P decreases. In con-trast, under N limitation, the NUE effect opposes the func-tional effect because the growth machinery has an Nconcentration lower than that of the uptake machinery,leading to increasing allocation to the growth machinery(RNA) even though growth rate is declining. However, atvery low resource levels, increased baseline biomass z willbe favored because of its even lower N concentration,despite the fact that this reduces both uptake and growthcapacity. Together, these responses provide a mechanisticexplanation for couplings between growth and biomassstoichiometry that lead to the non-GRH-compliant rela-tionships observed under N limitation (fig. 2).

Under C limitation, there is an NUE effect analogousto but smaller than that under N limitation (becausegrowth machinery is lower in C than is uptake machinery).However, under C limitation, maintenance respiration be-comes an important determinate of biomass C at a lowgrowth rate, which limits the importance of the NUE effectand maintains GRH-compliant relationships except at verylow growth rates (fig. 2).

Mechanisms Regulating Potential Suboptimal Variabilityin Biomass Composition

The modeled differences in the near-optimal ranges (thedistance between the thin lines in fig. 2) under limitationby different nutrients suggests that the relationship be-tween optimized g and the specific growth rate (and there-fore the RNA-G relationship) is much less constrained andmore prone to variation at low G under C limitation andunder N limitation than under P limitation. For example,the near-optimal range of RNA : protein under N limi-tation is ∼10 times that under P limitation at a low growthrate for Pectobacterium carotovorum (fig. 2). This difference

Bacterial Stoichiometry E37

Figure 4: Simulated C, N, and P recycling (EC, EN, and EP) and bacterial (Pectobacterium carotovorum) biomass element ratios in responseto external resource C : N ratio. The difference between homeostatic structural biomass (thin lines) and dynamically optimized biomasspartitioning (thick lines) was tested. Resource C : N was varied by reducing C availability from right to left on the X-axis, which caused ashift from N limitation to C limitation of bacterial growth (P was not limiting). The approximate shifting points are indicated by verticaldashed lines. a, b, Recycling (E) of C (dash-dotted lines), N (#5; dashed lines), and P (#5; solid lines). c, d, Structural biomass elementratios for C : N (solid line) and N : P (dashed line). a and c correspond to double the resource N level in b and d.

is due to the interaction of the NUE effect and the func-tional effect on the optimal biomass allocation. Under Plimitation these mechanisms work together, amplifying theeffect of P use on growth, whereas under N limitation theyoppose each other and therefore tend to cancel out. Thisdifference leads to a high sensitivity of G to P use andtherefore to variation in biomass composition (g and z)under P limitation, while under N limitation the corre-sponding sensitivity of G to biomass composition is low.

Novel Aspects and Implications forMicrobial Biomass Stoichiometry

Our model links physiology directly to biomass compo-sition and therefore to biomass stoichiometry, employingoptimization of structure composition as the controllingmechanism. We have shown that the optimization strategy,rooted in an ecological and evolutionary rationale, pro-vides a means to understand how microbial biomass stoi-chiometry and physiology are influenced by changes inthe environment, including resource variation. In additionto the novel optimization strategy, our model differs fromwidely used microbial stoichiometry models (e.g., quota

models: Droop 1968; Thingstad 1987) in its more mech-anistic representation of the growth process, which de-scribes gradual shifts between different resource limita-tions and the explicit interaction of growth machinery andmetabolite pools. Supported by empirical data, we showedhow this interaction leads to a positive concave relation-ship between G and the metabolite pool for a limitingnutrient, while the metabolite pool tends to decrease withG at high G for nonlimiting nutrients (fig. 3). Our modelprovides a mechanistic explanation for such relationships,which also have been observed for different nutrients inyeast (Boer et al. 2010).

We applied the model to elucidate observed patterns ofcovariation among biomass composition, stoichiometry,and growth, such as the growth-rate hypothesis (GRH),under different resource stoichiometry. Integration of stoi-chiometric theory, cellular composition, and physiologyhas previously been done on the basis of the GRH (Vredeet al. 2004) and by incorporating the GRH into the met-abolic theory of ecology (Allen and Gillooly 2009). How-ever, while the resulting theory explained patterns acrossorganism types, it did not explain how individual organ-isms and populations adapt or acclimate to changes in

E38 The American Naturalist

resource level and stoichiometry. By contrast, adaptivephenotypic plasticity, that is, optimal acclimation of bio-mass composition, is central in our model. Moreover, mul-tiple empirical studies show that a theory where ribosomecontent always correlates with growth seems insufficientfor dealing with microorganisms, because they often defythe GRH under non-P limitation (Flardh et al. 1992;Binder and Liu 1998; Elser et al. 2003). In the light of ourmodel, such results are explained by opposing effects ofbiomass composition on functional capacities versus nu-trient-use efficiency, exemplified for P. carotovorum underN limitation (fig. 2). In addition, at low G under C andparticularly N limitation, our model implies that G is rel-atively insensitive to variation in cell composition (fig. 2),suggesting an explanation for the high variability in stoi-chiometric relationships and RNA observed under theseconditions. Thus, our model offers a mechanistic expla-nation for observations of variable and high RNA abun-dance at low G, which has previously been suggested tobe a means of responding quickly to increased resourcelevels (Flardh et al. 1992). Our suggested mechanism mayalso have contributed to the decoupling of growth-rateRNA and biomass P observed in bacteria across a largenumber of lakes (Hall et al. 2009).

Although our results are consistent with the limited va-lidity of the GRH observed under many resource condi-tions, they do not imply that the GRH is exclusively re-stricted to P limitation, as suggested by Elser et al. (2003).We showed that organisms grown under C limitation(Escherichia coli; fig. 2) can give rise to a relationship be-tween G and RNA similar to that under P limitation. How-ever, in contrast to P limitation, under C limitation suchrelationships result from a purely functional trade-off be-tween uptake and growth capacity and are not affected bynutrient- (here C-) use efficiency (NUE). Although notshown, the functional trade-off also leads to GRH-com-pliant relationships under “balanced growth,” that is, whenall nutrients are colimiting, because there is no opposingeffect of NUE, as there is no benefit of increasing the NUEof any particular nutrient relative to other nutrients. Inagreement with these predictions, the GRH has been ver-ified for a range of microorganisms under balanced growthor C limitation (Karpinets et al. 2006).

The relationship between RNA and G varied signifi-cantly between E. coli and P. carotovorum (fig. 2), sup-porting empirical results that show that this relationshipis stronger within a species than across species (Kemp etal. 1993; Kerkhof and Kemp 1999) or communities (Hallet al. 2009). Furthermore, our model implies that thisrelationship is strongly linked to the capacity of the growthmachinery, that is, the maximal synthetic capacity of RNA(fG; eqq. [3]–[5]), which differs strongly between our in-vestigated species (table 2). This result explains mecha-

nistically why the synthetic capacity of rRNA is a key factorcontrolling differences in the RNA-G relationship acrossspecies with different ecological strategies, as has been pro-posed (Dethlefsen and Schmidt 2007). These interspeciesdifferences may also be reinforced by differences in theproportion of biomass not related to growth or uptake(z), which we predicted to be much higher for the slower-growing P. carotovorum than for the fast-growing E. coli.

Implications for Nutrient Cycling

Nutrient recycling (mineralization) is in large part a mi-crobial process that is strongly affected by the stoichi-ometry of microbial biomass (Cherif and Loreau 2007;Manzoni et al. 2008). Because our growth mechanism (eq.[3]) is based on the synthesizing-unit concept (Kooijman1998, 2001), it readily models the gradual shift from N toC limitation of microbial growth that gradually enhancesthe N (and P) mineralization : immobilization ratio duringthe decomposition process. Our results indicate that theexcretion of N and P during organic-matter decomposi-tion is increased if biomass is dynamically optimized rel-ative to excretion when biomass is strictly homeostatic.Therefore, our model results suggest that part of nutrientmineralization during organic-matter decomposition is aby-product of microorganisms adjusting their biomasscomposition to maximize growth (fig. 4). Thus, our sug-gested optimization mechanism may allow us to developa better understanding of nutrient mineralization and im-mobilization dynamics in ecosystems.

Limitations and Possibilities for Additional Evaluation

Our ability to validate all aspects of the model for a widerrange of microorganisms was limited by the availability ofdata that included all necessary biomass compartments,growth rate, and a conclusive identification of the limitingnutrient. In particular, direct evidence for changes in up-take capacity would be valuable for additional testing ofthe model and the potential effects on nutrient recycling.However, while the link between growth machinery andRNA is well established, a measure of uptake machineryis not readily obtainable from macromolecular composi-tion. The application of stable-isotope probing and single-cell techniques to microbial ecology may allow for higherresolution and more specific estimates of uptake dynamicsin future studies (Wagner 2009).

Although our data support our assumption that themetabolite pools are in equilibrium with the growth rate(fig. 3), this may not always be the case, for example, underrapid variations in resource level. However, if the equilib-rium assumption (eq. [2]) is relaxed, the model should

Bacterial Stoichiometry E39

also be valid under nonequilibrium conditions, whichwould be an interesting topic for further analysis.

Conclusions

Our analysis suggests that maximization of specific growthrate G, constrained by a trade-off between growth anduptake capacity, is a dominant control on bacterial biomasscomposition. In addition, the shifts in biomass compo-sition affect the resource demands for a given growth rate,that is, nutrient-use efficiency, because the elemental com-position of each cellular component is unique. These twomechanisms can be complementary, as under P limitation,and can strengthen the relationship between growth andbiomass stoichiometry. They can also act antagonistically,for example, under N limitation. At a low resource level,where nutrient-use efficiency becomes a more importantcontrol than the functional growth-uptake trade-off, thenutrient-use efficiency optimization will lead to differentbiomass allocation patterns, depending on which nutrientis limiting. For example, the growth-rate hypothesis(GRH) is valid under P limitation but breaks down underN limitation. The elucidation of these specific mechanismsprovides the first clear mechanistic explanation for whyGRH relationships can come uncoupled under certain re-source stoichiometry conditions. In addition, the modelresults suggest that optimization of cellular compositionduring decomposition of organic matter indirectly in-creases N and P mineralization. Such dynamics may playan important role in nutrient cycling in ecosystems.

The ability of the model to mechanistically predict bac-terial biomass stoichiometry and the model’s potentialimplications for nutrient recycling suggest that its mech-anisms may be an appropriate starting point for incor-porating dynamic microbial physiology into ecosystem-level models. Because our model is rooted in an ecologicaland evolutionary rationale, it can be used to mechanis-tically explain patterns of microbial stoichiometry in anecological context. In addition to established empirical pat-terns of what happens (e.g., biomass P : N ratio increaseswith G), the model explains when it happens (e.g., underC limitation), how it happens (e.g., a growth-uptake trade-off), and why it happens (ecological optimization).

Acknowledgments

This work is a contribution from the Austrian nationalresearch network MICDIF (Linking Microbial Diversity

and Functions across Scales and Ecosystems) and was sup-ported by the Austrian Science Foundation (Fonds zurForderung der wissenschaftlichen Forschung, grantS10008-BI7) and IIASA, the International Institute of Ap-plied Systems Analysis.

APPENDIX A

The Synthesizing Unit (SU) and Growth

An SU is analogous to an enzyme to which substrate mol-ecules bind to form a product that is then released (Kooij-man 1998, 2001). In our case, the product is biomass andthe substrates are carbon (C), nitrogen (N), and phos-phorus (P); all other atoms in biomass and substrate areimplicit. The SU has binding sites for all substrate mol-ecules, such that all sites must bind substrate before theproduct is released. The production rate of the SU dependson the probabilities per time unit that the substrate mol-ecules bind (substrate flux) relative to the number of bind-ing sites. In our model, the relative number of bindingsites among elements in the SU mirrors the content of theelements in biomass (bX). The substrate flux (bindingprobability) is proportional to the metabolite concentra-tions for each nutrient (pX). This formulation is based onan SU model simplified for production of a generalizedcompound (biomass) limited by three substrates (C, N,and P; Kooijman 1998).

The above describes the growth properties per unitgrowth machinery (corresponding to 1 SU). To incorpo-rate the effects of variable amounts of growth machinery(many SUs), we apply basic Michaelis-Menten-type ki-netics. At low amounts of growth machinery relative to p,specific growth (G) is limited by Gmax (eq. [4]), which isproportional to the amount of growth machinery g (1 �) and its maximum capacity for biomass synthesis (fG),z

which is a function of the maximal translational activityof the ribosomes (see Jackson et al. 2008). At high amountsof growth machinery relative to metabolite concentration,many units of growth machinery compete for little sub-strate, so that G is not limited by the amount of growthmachinery but by its efficiency (eg), that is, the productformation of each SU per nutrient flux.

E40 The American Naturalist

APPENDIX B

Modeled Growth and Excretion

Figure B1: Modeled microbial growth (solid line) and excretion(dashed line) as functions of the internal nutrient metabolite pool(pN). Excretion prevents pN from exceeding the maximum p pN

(dotted vertical line).p lim N

APPENDIX C

Experiment for Pectobacterium carotovorum

Treatments

Pectobacterium carotovorum (Bergey et al. 1923, pp. 374–375) was cultivated in batch cultures, in three replicates,in liquid minimal media modified from the one used byClark and Maaløe (1967). The media contained 30 mMMOPS (pH 7.0), 0.1 nM CaCl2 0.1, 3 mM FeSO4, 20 mMKCl, 2 mM MgCl2, 14 mM Na2SO4, and 51 mM NaCl.Glucose, NH4Cl, and Na2HPO4 were added as carbon (C),nitrogen (N), and phosphorus (P) sources, respectively, indifferent amounts and ratios, giving a total of 12 com-binations (treatments) of C, N, and P. Bacterial media weresterilized by autoclaving at 121�C for 20 min. Glucose wasadded to autoclaved media from filter-sterilized (pore size:0.2 mm) stock solutions. The inoculations on the mediawere made from precultures grown for 60 h in the re-spective minimum media with fixed concentrations of 150mM C, 12.5 mM N, and 0.25 mM P. The precultures werethen centrifuged; the pellets were washed twice with min-imal media without C, N, or P sources and then resus-pended once more in minimal media to inoculate the 12treatments. Cultivation was performed in 400 mL of mediaat 22�C on a rotary shaker at 180 rpm. For determination

of biomass and its elemental composition, each of the 12treatments was run in 400 mL of either three or fourreplicates as described above. For each treatment, the bac-terial cultures were measured at two growth phases, log-arithmic (initial fast growth) and stationary (late slow,strongly resource-limited growth).

Measurements

The bacterial cultures were transferred into centrifugeflasks and centrifuged for 45 min at 10,845 g. The biomasspellet was washed twice with 50 mL of 130 mM NaCl,followed by resuspension in 50 mL of 130 mM NaCl andcentrifugation for 10 min at 10,845 g. The biomass wasdried in a drying oven for determination of dry mass andC, N, and P content. Aliquots of dried biomass (∼2 mg)were weighed into tin capsules, and total C and N weredetermined with an elemental analyzer (EA 1110, CE In-struments, Milan, Italy). For analysis of total P in biomass,approximately 10 mg of dried samples were wet digestedwith 1 mL of nitric acid–perchloric acid mixture (4 : 1 ratioof HNO3 to HClO3; Kolmer et al. 1951, pp. 1090–1091)in 2-mL glass flasks on a heating plate. For acid digestion,the temperature was increased stepwise to 250�C and thenmaintained until a small residual volume was left in theglass flask. Samples were cooled to room temperature andfilled to the 2-mL level with milliQ water. Inorganic P inthe digests was quantified photometrically on the basis ofthe phosphomolybdate blue reaction (Schinner et al. 1993)in a microtiter plate format with a microplate reader (BIO-TEK Instruments). RNA content was measured by fluo-rometry with the fluorescent stain RiboGreen accordingto the methods of Makino and Cotner (2004). The growthrates were determined by measuring the increase in tur-bidity. Turbidity was monitored at regular intervals witha microplate reader with 250-mL aliquots at a wavelengthof 450 nm for bacteria. Optical density (OD) growth ki-netics was constructed by plotting the OD of suspensionscorrected for noninoculated medium versus time of in-cubation. At biomass densities higher than , theOD 1 0.6samples were diluted.

Calculation of Growth Rate and Identification ofLimiting Resource

The bacterial growth rate was determined by using a five-parameter logistic growth model that was fitted (using theLevenberg-Marquardt algorithm in the software Sigma-Plot) to measured OD as a function of time. Specificgrowth rate (G) was evaluated for each biomass samplingpoint ts as growth in OD divided by OD; that is, G p

at . Observations at extremely low[(dOD/dt)/OD] t p ts

Bacterial Stoichiometry E41

growth rates not significantly different from 0 were excludedbecause of the high uncertainty in the growth-rate estimate.

The change in non–nucleic acid P (P � P �total RNA

) with growth rate be-P ≈ nonstructural, metabolite PDNA

tween the two growth phases was used to detect the lim-iting resource. According to our theory, P limitation wasindicated by decreased non–nucleic acid P concentrationduring the transition from initial (fast) to late (slow)growth. The results were also confirmed by an analysis ofpolyphosphate (indicative of P storage). Absence of P lim-itation was interpreted as N limitation. Carbon limitationwas excluded because carbohydrate concentration (indi-cating C storage) increased between the logarithmic andstationary phases in all treatments.

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Associate Editor: Volker GrimmEditor: Mark A. McPeek

Transmission electron microscopy image of bacteria (Pectobacterium carotovorum) used in the study. The dark blue centers of the cellscontain the cellular machinery and pools of storage compounds, which both adapt to the external resource availability. Photograph byEdward K. Hall.